Discrete Fourier Series and Transforms

Transcription

1 Lctur 4 Outi: Discrt Fourir Sris ad Trasforms Aoucmts: H 4 postd, du Tus May 8 at 4:3pm. o at Hs as soutios wi b avaiab immdiaty. Midtrm dtais o t pag H 5 wi b postd Fri May, du foowig Fri (as usua) Rviw of Last Lctur Discrt Fourir Sris Discrt Fourir Trasform Ratio btw DFT & DTFT DFT as a Matri Opratio Proprtis of DFS ad DFT Circuar Tim/Frq. Shift ad Covoutio

2 Midtrm Dtais Tim/Locatio: Friday, May, :32:5pm i this room. Op boo ad ots you ca brig ay writt matria you wish to th am. Cacuators ad ctroic dvics ot aowd. i covr a cass matria from Lcturs 3. Practic MT postd today, worth 25 tra crdit poits for taig it (ot gradd). Ca b turd i ay tim up uti you ta th am Soutios giv wh you tur i your aswrs I additio to practic MT, w wi aso provid additioa practic probms/sos MT Rviw i cass May 7 Discussio Sctio May 8, 4:36 (MT rviw ad practic probms) Rguar OHs for m/tas this w ad t (o w H t w) I am aso avaiab by appoitmt

3 Rviw of Last Lctur FIR dsig tais choic of widow fuctio to mitigat Gibbs Goa is to approimatd dsird fitr without Gibbs/wiggs Dsig tradoffs ivov mai ob vs. sidob sizs Typica widows: rctag (bocar), triag, Haig, ad Hammig FIR dsig for dsird h d [] tais picig a gth M, sttig h a []h d [], M/2, choosig widow w[] with h w []h[]w[]to mitigat Gibbs, ad sttig h[]h w [M/2] to ma dsig causa FIR impmtd dircty usig M day mts ad M+ mutipirs Ca itroduc group day Efficity impmtd with DFT Eamp dsig for LFP (Diffrtiator i H) Hammig smooths out wiggs from rctaguar widow Itroducs mor distortio at trasitio frqucis tha rctaguar widow

4 Discrt Fourir Sris Th DFS is th DTFS with a diffrt ormaizatio: Cosidr a priodic discrttim siga ~ :. Usi Th a is aso priodic: Dfi [ + ] ~ ~ ~ 2p j Appars i g th DTFS, DFS or DFT for priodic squcs this 2p ± j ota tio, w writ Th w ca writ ~ ~ [ + ] Usig this otatio, w hav th DFS pair for priodic sigas: å ~ ~ ~ a Simp computatio of mas this pair asir to comput tha DTFS ~ j (2 p / ) å < > å ~

5 Discrt Fourir Trasform (DFT) ors with oy o priod of ì ~ í î othrwis ad Ca rcovr origia priodic squcs ~, ~ as ~ [ (( ) ) ] [ r] ~ (( ) ) mod r å r Equivaty, wor with samps of [] Lads to DFT Pair å Ivrs DFT ~ ì ~ í î othrwis [(( )) ] å r å DFT ~ (( ) ) mod Cojugat Ratioships * { } DFT { } ( ) * DFT/IDFT commoy usd i DSP, usig gth siga bocs, du to its much owr computatioa compity tha th DTFT/IDTFT DFT ( { }) * * { } DFT DFT

6 Eamp Ra ad v ad ~ ~ ~ ~ Opriod rprstatios:

7 Ratio btw DFT ad DTFT Giv a gth siga [], poit DFT is Its DTFT is DFT is th DTFT sampd at quay spacd frqucis btw ad 2p:, 2 å å j p ( ) å j j ( ), 2 j p or

8 DFT/IDFT as Matri Opratio DFT Ivrs DFT Computatioa Compity Computatio of a poit DFT or ivrs DFT rquirs 2 comp mutipicatios. ( ) ( ) ( ) ( ) ( )( ) ø ö è æ ø ö è æ ø ö è æ ( ) ( ) ( ) ( ) ( )( ) ø ö è æ ø ö è æ ø ö è æ

9 Proprtis of th DFS/DFT

10 Circuar Tim/Frqucy Shift Circuar Tim Shift (provd by DFS proprty of ~ ) j m m [(( m) ) ]«Circuar Frqucy Shift (IDFS proprty of ) 2p 2p j «[(( ) ) ] ~

11 Circuar Covoutio Dfid for two gth squcs as ì ï ~ [ ] å å ~ m 2 m 2 ím m ïî othrwis [ m] [(( m) ) ] 2 Circuar covoutio i tim is mutipicatio i frqucy å m DFT [ m] [(( m) ) ] 2 «2 Duaity «2 2

12 Mai Poits DFS is th DTFS with a diffrt ormaizatio DFT oprats o o gth pic of a siga [] Fast/ow compity computatio ( 2 comp mutipicatios) DFT is th DTFT sampd at quay spacd frqucis btw ad 2p DFT/IDFT ca b cacuatd via a matri mutipicatio DFS/DFT hav simiar proprtis as DTFS/DTFT but with modificatios du to priodic/circuar charactristics